Lemma 21.53.3 (09BE)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.53: Complexes with locally constant cohomology sheaves
Lemma 21.53.3 (cite)

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Lemma 21.53.3. Let $\mathcal{C}$ be a site with final object $X$ . Let $\Lambda$ be a ring. Let $K, L$ be objects of $D(\Lambda )$ with $K$ perfect. Let $\varphi : \underline{K} \to \underline{L}$ be map in $D(\mathcal{C}, \Lambda )$ . There exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is equal to $\underline{\alpha _ i}$ for some map $\alpha _ i : K \to L$ in $D(\Lambda )$ .

Proof. Follows from Lemma 21.53.2 and Modules on Sites, Lemma 18.43.3. $\square$

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